Non-Gaussian operations on bosonic modes of light: Photon-added Gaussian channels

We present a framework for studying bosonic non-Gaussian channels of continuous-variable systems. Our emphasis is on a class of channels that we call photon-added Gaussian channels, which are experimentally viable with current quantum-optical technologies. A strong motivation for considering these channels is the fact that it is compulsory to go beyond the Gaussian domain for numerous tasks in continuous-variable quantum information processing such as entanglement distillation from Gaussian states and universal quantum computation. The single-mode photon-added channels we consider are obtained by using two-mode beam splitters and squeezing operators with photon addition applied to the ancilla ports giving rise to families of non-Gaussian channels. For each such channel, we derive its operator-sum representation, indispensable in the present context. We observe that these channels are Fock preserving (coherence nongenerating). We then report two examples of activation using our scheme of photon addition, that of quantum-optical nonclassicality at outputs of channels that would otherwise output only classical states and of both the quantum and private communication capacities, hinting at far-reaching applications for quantum-optical communication. Further, we see that noisy Gaussian channels can be expressed as a convex mixture of these non-Gaussian channels. We also present other physical and information-theoretic properties of these channels.

Figure 1

Schematic diagram for the construction of a class of non-Gaussian channels using two constituent elements of photon addition and Gaussian unitary gates.

Figure 2

Unitary dilation for Gaussian channels. Any quantum-limited single-mode bosonic Gaussian channel can be realized in the following three-step method: First one appends a suitable single-mode environment state that can be taken to be pure Gaussian (vacuum, without loss of generality), then evolves the system-environment state through a canonical two-mode Gaussian unitary, and finally, traces out the environment degrees of freedom.

Figure 3

Basic schemes to construct non-Gaussian channels using the Stinespring representation. One can use an ancilla non-Gaussian state $|{\psi }_{\mathrm{nonG}}\rangle$ and a Gaussian unitary as in scheme 1 (top), which we label Gaussian dilation scheme. The second option is to use a non-Gaussian unitary with a Gaussian ancilla as in scheme 2 (bottom), which we label the general dilation scheme. A third scheme that combines the use of a non-Gaussian unitary and a non-Gaussian ancilla state can be reduced to scheme 2. In this article we focus on scheme 1.

Figure 4

Scheme for obtaining photon-added Gaussian channels. Starting from the underlying unitary representation for quantum-limited attenuator, amplifier, and phase-conjugation channels, we obtain the corresponding photon-added version by applying photon addition to the vacuum state to generate a Fock state in the environment. The resulting channels fall under the class of linear bosonic non-Gaussian channels. In the absence of photon addition we have the underlying single-mode quantum-limited bosonic Gaussian channel.

Figure 5

Realization of the photon-added amplifier channels $\mathcal{A}(\kappa ;n)$. Here $\kappa$ denotes the channel parameter coming from the two-mode squeezing (TMS) operator unitary and $n$ denotes the photon addition.

Figure 6

Structure of Kraus operators for the photon-added amplifier channels. Each Kraus operator has nonzero elements only along a line parallel to the diagonal whose location depends on $|\ell -n|$, where $\ell$ is the Kraus index and $n$ the number of photon addition.

Figure 7

Realization of the photon-added attenuator channels $\mathcal{B}(\kappa ;n)$. Here $\kappa$ denotes the channel parameter coming from the two-mode beam-splitter unitary and $n$ denotes the photon addition.

Figure 8

Structure of Kraus operators for the photon-added attenuator channels. Each Kraus operator with index $\ell$ has nonzero elements only along a line parallel to the diagonal, whose location depends on $|\ell -n|$, where $n$ is the photon-number state of the environment.

Figure 9

Realization of the photon-added amplifier channels $\mathcal{C}(\kappa ;n)$. Here $\kappa$ denotes the channel parameter coming from the two-mode phase-conjugation unitary (PCU) and $n$ denotes the photon addition.

Figure 10

Structure of Kraus operators for the photon-added phase-conjugation channels. Each Kraus operator has nonzero elements only along a particular antidiagonal whose location depends on $\ell +n$, where $\ell$ is the Kraus index and $n$ is the value of photon addition.

Figure 11

Relation between the unitary operators in the Stinespring dilation of the amplifier channel denoted by TMS and that of the phase-conjugation channel with parameters $\kappa$ and ${\kappa }^{\prime }=\sqrt{{\kappa }^2-1}$ (top), and the two unitaries (beam splitters) in the Stinespring dilation of the corresponding attenuator channels with parameters $\kappa$ and ${\kappa }^{\prime }=\sqrt{1-{\kappa }^2}$ (bottom), respectively. The pairwise unitaries in each case are connected by the mode-flip operator.

Figure 12

Coherent information $I_{\mathrm{coh}}(\mathcal{C}(\kappa ;n),{\widehat{\rho }}_{})$ as a function of photon addition $n$ with $\kappa =\sqrt{1.5^2-1}$ and ${\widehat{\rho }}_{}=0.6|0\rangle \langle 0|+0.4|1\rangle \langle 1|$. The coherent information is positive for $n$ varying from 1 to 10.

Figure 13

Variation of entropy as a function of the truncated value of the output Fock state. The top line (in blue with circle plot points) corresponds to $S(\mathcal{C}(\kappa ;1)\left[{\widehat{\rho }}_{}\right])$ and $\kappa =\sqrt{1.5^2-1}$ and the bottom line (in red with square plot points) corresponds to $S(\mathcal{A}(1.5;1)\left[{\widehat{\rho }}_{}\right])$, with ${\widehat{\rho }}_{}=0.6|0\rangle \langle 0|+0.4|1\rangle \langle 1|$.

Figure 14

Variation of coherent information $I_{\mathrm{coh}}(\mathcal{C}(\kappa ;1),{\widehat{\rho }}_{\mathrm{in}})$ as a function of the channel parameter $\kappa$ with ${\widehat{\rho }}_{\mathrm{in}}=0.6|0\rangle \langle 0|+0.4|1\rangle \langle 1|$ and $n=1$.

Figure 15

Scheme for environment-assisted error correction for transmitting classical information through the photon-added attenuator, amplifier, and phase-conjugation channels. Here the label $x$ stands for classical information encoded into Fock states and input to the channel. Conditioned on classical feedback $y$ from the environment that results from a measurement, a recovery map $R_y$ is applied at the output of the channel to achieve error correction.

Figure 16

Schematic for photon-added noisy Gaussian channels with a vacuum environment state. In the absence of photon addition we obtain the underlying noisy Gaussian channel with the noise parameter tuned by the squeezing parameter of the TMS operator.

Figure 17

Schematic for photon-added noisy Gaussian channels with a mixed environment state. Here PATS stands for the photon-added thermal state.